A Comparison Between Lie Group- and Lie Algebra- Based Potential Functions for Geometric Impedance Control

TL;DR

The paper analyzes two geometric impedance control formulations on SE(3): a Lie group-based method using a Frobenius-distance-based potential and a Lie algebra-based method using a log-map potential. By deriving distance metrics, potential functions, and dissipative control laws, the authors establish both qualitative and quantitative comparisons, including stability analyses and numerical experiments on a UR5e manipulator. The Lie algebra-based approach yields stronger convergence properties (almost global exponential stability) and tighter handling of large initial errors, while the Lie group-based approach offers simplicity and natural separation of translational and rotational components. The work highlights that both methods preserve SE(3) equivariance, making them suitable for learning-based strategies and transfer across tasks, with practical implications for robust manipulation in interactive environments.

Abstract

In this paper, a comparison analysis between geometric impedance controls (GICs) derived from two different potential functions on SE(3) for robotic manipulators is presented. The first potential function is defined on the Lie group, utilizing the Frobenius norm of the configuration error matrix. The second potential function is defined utilizing the Lie algebra, i.e., log-map of the configuration error. Using a differential geometric approach, the detailed derivation of the distance metric and potential function on SE(3) is introduced. The GIC laws are respectively derived from the two potential functions, followed by extensive comparison analyses. In the qualitative analysis, the properties of the error function and control laws are analyzed, while the performances of the controllers are quantitatively compared using numerical simulation.

Figures (6)

• Figure 1: The concept behind the definition of the velocity error. (a) $\dot{g}_d \in T_{g_d}G$ is first translated via right multiplication $g_d^{-1}g$. (b) $g_{de}$ as a whole is interpreted as a group element and the vector $\dot{g}_{de} \in T_{g_{de}}G$ is analyzed directly.
• Figure 2: Comparison of Lie group and Lie algebra-based approaches for (Left) Error functions $\Psi_1(R,I)$ and $\Psi_2(R,I)$, (Right) Elastic forces $f_{_{G,1}}(R,I)$ and $f_{_{G,2}}(R,I)$ with identity gains.
• Figure 3: The regulation results with large initial errors in $x$, $y$, and $z$ coordinates are plotted for $\text{GIC-1}$ in red solid lines and $\text{GIC-2}$ in blue dashed lines.
• Figure 4: Error function \ref{['eq:error_fun_SE(3)']}$\Psi_1(g,g_d)$ for the resulting trajectories.
• Figure 5: The trajectory tracking results with smooth desired trajectories in $x$, $y$, and $z$ coordinates are plotted for $\text{GIC-1}$ in red solid lines and $\text{GIC-2}$ in blue dashed lines. All lines are on top of each other and indistinguishable.

Theorems & Definitions (6)

• Remark 1
• Theorem 1
• proof
• Theorem 2
• proof
• Remark 2